By expanding in appropriate series of singular harmonic functions a solution is found to the equipotential problem for four circular poles (or electrodes) with quadrupole symmetry. The duodecapole term in the potential is found to be absent when the ratio of pole diameter to throat diameter is 1.14511; the contributions from multipoles of higher order are found to be small. The method is extended to permit the calculation of the effects of placing the exciting coils near the poles, of allowing the permeability to be finite (but uniform), and of adding a yoke. THe distribution of flux density on a pole surface is calculated. It is recommended that circular arcs replace hyperbolic arcs as the basis for the design of quadrupole lenses.